Radial Symmetry Does Not Preclude Condorcet Cycles If Different Voters Weight the Issues Differently

Abstract

Radial symmetry, by our definition, is a precise condition on continuous ideal-point distributions, rarely if ever found exactly in practice, that is similar to the classical 1967 symmetry condition of Plott but pertains to an infinite electorate; the bivariate normal distribution provides an example. A Condorcet cycle exists if the electorate prefers alternative X to Y, Y to Z, and Z to X. An alternative K is a Condorcet winner if there is no alternative that the electorate prefers to K. Lack of a Condorcet winner may engender turmoil. The nonexistence of a Condorcet winner implies that a Condorcet cycle exists. Radial symmetry precludes the existence of Condorcet cycles and thus guarantees a Condorcet winner; but this result assumes that all voters weight the dimensions alike. Our counterexamples show that a Condorcet cycle can arise, even under radial symmetry, if the weighting of issues varies across voters. This finding may be of more than theoretical value: It may suggest that in an empirical setting (without radial symmetry), a Condorcet cycle may be more frequent if voters differ as to how they weight the dimensions. We examine, for illustration based on two dimensions (left–right, linguistic), a Condorcet preference cycle in Finland’s 1931 presidential election.